Antiderivatives and the Fundamental Theorem of Calculus

Antiderivatives and the Fundamental Theorem of Calculus

Antiderivatives and the Fundamental Theorem of Calculus

In this section we present the fundamental theorem of calculus. First, we must make
a definition. A function
F(x)
is called an antiderivative of a function
f (x)
if
f (x)
is the derivative of
F(x)
; that is, if
F'(x) = f (x)
. The antiderivative of
a function
f (x)
is not unique, since adding a constant to a function does not change
the value of its derivative:

(f (x) + c) = (f (x)) + (c) =

It is true, however, that any two antiderivatives of a function
f (x)
will differ by a
constant.

Why are antiderivatives a useful concept? Because they provide a shortcut for
calculating definite integrals, as shown by the first part of the fundamental theorem of
calculus.

Fundamental Theorem of Calculus 1

Let
f (x)
be a function that is integrable on the interval
[a, b]
and let
F(x)
be an
antiderivative of
f (x)
(that is,
F'(x) = f (x)
). Then

f (x)dx = F(b) - F(a)

Since the expression
F(b) - F(a)
is one we will encounter often, we will sometimes
employ a special shorthand to simplify our equations:

F(x)|ab = F(b) - F(a)

Note that any antiderivative
F(x)
will give the same value for
F(b) - F(a)
, since any
two antiderivatives differ only by a constant, which will cancel upon subtraction. The
formula in the theorem may also be written